Practical methods to estimate horizontal and vertical permeabilities

ABSTRACT

In one method, the permeabilities are obtained by correcting the geometric factor derived from combining the FRA analysis and buildup analysis.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 11/014,422 filed on Dec. 16, 2004, which claimed priority from U.S. Provisional Patent Application Ser. No. 60/604,552 filed on 26 Aug. 2004, the contents of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention is related to the field of instruments used to sample fluids contained in the pore spaces of earth formations. More specifically, the invention is related to methods of determining hydraulic properties of anisotropic earth formations by interpreting fluid pressure and flow rate measurements made by such instruments.

2. Description of the Related Art

Electric wireline formation testing instruments are used to withdraw samples of fluids contained within the pore spaces of earth formations and to make measurements of fluid pressures within the earth formations. Calculations made from these pressure measurements and measurements of the withdrawal rate can be used to assist in estimating the total fluid content within a particular earth formation.

A typical electric wireline formation testing instrument is described, for example, in U.S. Pat. No. 5,377,755 issued to Michaels et al. Electric wireline formation testing instruments are typically lowered into a wellbore penetrating the earth formations at one end of an armored electrical cable. The formation testing instrument usually comprises a tubular probe which is extended from the instrument housing and then is impressed onto the wall of the wellbore. The probe is usually sealed on its outside diameter by an elastomeric seal or packing element to exclude fluids from within the wellbore itself from entering the interior of the probe, when fluids are withdrawn from the earth formation through the probe. The probe is selectively placed in hydraulic communication, by means of various valves, with sampling chambers included in the instrument. Hydraulic lines which connect the probe to the various sample chambers can include connection to a highly accurate pressure sensor to measure the fluid pressure within the hydraulic lines. Other sensors in the instrument can make measurements related to the volume of fluid which has entered some of the sample chambers during a test of a particular earth formation. U.S. Pat. No. 6,478,096 to Jones et al. discloses a formation pressure tester that is part of a bottomhole assembly used in drilling and can make measurements while drilling (MWD).

Properties of the earth formation which can be determined using measurements made by the wireline formation testing instrument include permeability of the formation and static reservoir pressure. Permeability is determined by, among other methods, calculating a rate at which a fluid having a known viscosity moves through the pore spaces within the formation when a predetermined differential pressure is applied to the formation. As previously stated, the formation testing instrument typically includes a sensor to make measurements related to the volume of fluid entering the sample chamber, and further includes a pressure sensor which can be used to determine the fluid pressure in the hydraulic lines connecting the probe to the sample chamber. It is further possible to determine the viscosity of the fluid in the earth formation by laboratory analysis of a sample of the fluid which is recovered from the sample chamber.

The permeability of a reservoir is an important quantity to know as it is one of the important factors determining the rate at which hydrocarbons can be produced from the reservoir. Historically, two types of measurements have been used for determination of permeability. In the so-called drawdown method, a probe on a downhole tool in a borehole is set against the formation. A measured volume of fluid is then withdrawn from the formation through the probe. The test continues with a buildup period during which the pressure is monitored. The pressure measurements may continue until equilibrium pressure is reached (at the reservoir pressure). Analysis of the pressure buildup using knowledge of the volume of withdrawn fluid makes it possible to determine a permeability.

In the so-called buildup method, fluid is withdrawn from the reservoir using a probe and the flow of fluid is terminated. The subsequent buildup in pressure is measured and from analysis of the pressure, a formation permeability is determined.

U.S. Pat. No. 5,708,204 to Kasap having the same assignee as the present application and the contents of which are fully incorporated herein by reference, teaches the Fluid Rate Analysis (FRA) method in which data from a combination of drawdown and buildup measurements are used to determine a formation permeability.

The methods described above give a single value of permeability. In reality, the permeability of earth formations is anisotropic. It is not uncommon for horizontal permeabilities to be ten or more times greater than the vertical permeability. Knowledge of both horizontal and vertical permeabilities is important for at least two reasons. First, the horizontal permeability is a better indicator of the productivity of a reservoir than an average permeability determined by the methods discussed above. Secondly, the vertical permeability provides useful information to the production engineer of possible flow rates between different zones of a reservoir, information that is helpful in the setting of packers and of perforating casing in a well. It is to be noted that the terms “horizontal” and “vertical” as used in the present document generally refers to directions in which the permeability is a maximum and a minimum respectively. These are commonly, but not necessarily horizontal and vertical in an earth reference frame. Similarly, the term “horizontal” in connection with a borehole is one in which the borehole axis is parallel to a plane defined by the horizontal permeability.

U.S. Pat. No. 4,890,487 to Dussan et al. teaches a method for determining the horizontal and vertical permeabilities of a formation using measurements made with a single probe. The analysis is based on representing the fluid behavior during drawdown by an equation of the form:

$\begin{matrix} {{{P_{f} - P_{i}} = \left( {\frac{Q\;\mu}{2\pi\; r_{p}k_{h}}{F\left( {\frac{\pi}{2},\sqrt{1 - {k_{V}/k_{H}}}} \right)}} \right)},} & (1) \end{matrix}$ where

-   P_(f) represents pressure of the undisturbed formation; -   P_(i) represents pressure at the end of draw-down period i; -   Q_(i) represents volumetric flow rate during draw-down period i; -   μ represents dynamic viscosity of the formation fluid; -   r_(p) represents the probe aperture radius; -   k_(H) represents horizontal formation permeability; -   k_(V) represents vertical formation permeability; and -   F denotes the complete elliptic integral of the first kind.     In Dussan, at least three sets of measurements are made, such as two     drawdown measurements and one buildup measurement, and results from     these are combined with a table lookup to give an estimate of     vertical and horizontal permeability. The above equation was derived     based on several assumptions: an infinite wellbore, constant     drawdown rate and steady state flow. The steady state flow condition     cannot be satisfied in a low permeability formation, or unless a     long test time is used. A constant drawdown rate is not reachable in     practice because the tool needs time for acceleration and     deceleration. The storage effect also makes it difficult to reach a     constant drawdown rate. The infinite wellbore assumption excludes     the wellbore effect on the non-spherical flow pattern, making their     method not inapplicable to high k_(H)/k_(V) cases. The cases of     k_(H)/k_(V)<1 were not presented in Dussan. The method works only in     a homogeneous formation. However, their method does not have any     procedure to check if the condition of homogeneous formation can be     satisfied for a real probe test. The present invention addresses all     of these limitations.

U.S. Pat. No. 5,265,015 to Auzerais et al. teaches determination of vertical and horizontal permeabilities using a special type of probe with an elongate cross-section, such as elliptic or rectangular. Measurements are made with two orientations of the probe, one with the axis of elongation parallel vertical, and one with the axis of elongation horizontal. The method requires a special tool configuration. To the best of our knowledge, there does not exist such a tool and it is probably difficult or expensive to build one. The present invention does not require a special tool, and such tool is available, for example, the one described in U.S. Pat. No. 6,478,096 to Jones et al.

U.S. Pat. No. 5,703,286 to Proett et al. teaches the determination of formation permeability by matching the pressure drawdown and buildup test data (possibly over many cycles). There is a suggestion that the method could be modified to deal with anisotropy and explicit equations are given for the use of multiple probes. However, there is no teaching on how to determine formation anisotropy from measurements made with a single probe. Based on the one equation given by Proett, it would be impossible to determine two parameters with measurements from a single probe. It would be desirable to have a method of determination of anisotropic permeabilities using a single probe. The present invention satisfies this need.

SUMMARY OF THE INVENTION

One embodiment disclosed herein is a method of estimating the permeability of an earth formation containing a fluid. Transient pressure measurements are made with the probe in a borehole in the earth formation. The probe has a substantially circular aperture and is in hydraulic communication with the earth formation. First and second estimates of permeability are obtained from the pressure measurements. From the first and second estimates of permeability, a horizontal permeability and/or a vertical permeability of the formation is determined. The pressure measurements may include a drawdown involving withdrawal of the fluid from the earth formation, and stopping the withdrawal of the fluid at a defined time. The first estimate of permeability may be based on analyzing a first set of pressure measurements made after the defined time and a second estimate of permeability may be based on analyzing a second set of pressure measurements made before the defined time. The method may further involve determining a ratio of a horizontal permeability and a vertical permeability of the formation and/or a spherical permeability of the formation. The permeability determination may be based on using a geometric factor relating to anisotropy and/or a geometric skin factor relating to non-spherical fluid flow. The second estimate of permeability may further involve matching pressure measurements made before and after the defined time using the first permeability and by adjusting a geometric skin factor.

Another embodiment disclosed herein is an apparatus for evaluating an earth formation which contains a fluid. The apparatus includes a probe having a substantially circular aperture configured to be conveyed in a borehole and in hydraulic communication with the earth formation. The apparatus includes a pressure sensing device configured to make transient pressure measurements in the probe. The apparatus further includes a processor configured to estimate a first permeability and a second permeability from the pressure measurements, and determine from the first and second permeability a horizontal permeability of the formation and/or a vertical permeability of the formation. The pressure measurements may further include a drawdown involving the withdrawal of fluid from the earth formation and stopping the withdrawal of the fluid at a defined time. A processor is configured to obtain the first estimate of permeability at least in part by analyzing a first set of pressure measurements made after the defined time, and obtaining the second estimate of permeability at least in part by analyzing pressure measurements made before and after the defined time. The processor may further be configured to determine a ratio of horizontal permeability and vertical permeability and/or spherical permeability of the formation. The apparatus may further include a conveyance device configured to convey the probe into the borehole. The conveyance device may be a wireline or a drilling tubular.

Another embodiment disclosed herein is a computer-readable medium for use with an apparatus for evaluating an earth formation containing a formation fluid. The apparatus includes a probe having a substantially circular aperture configured to be conveyed in a borehole. A pressure sensing device is configured to make transient pressure measurements in the probe. The medium includes instructions which enable a processor to estimate a first permeability and a second permeability from the pressure measurements, and determine from the first permeability and a second permeability a horizontal permeability of the formation and/or a vertical permeability of the formation. The medium may include a ROM, an EPROM, an EEPROM, a Flash Memory, and an optical disk.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is best understood with reference to the accompanying figures in which like numerals refer to like elements and in which:

FIG. 1 (prior art) is an illustration of a wireline conveyed formation testing instrument positioned within a wellbore;

FIG. 2 (prior art) shows a graph of measured pressure with respect to fluid flow rate in the earth formation;

FIG. 3 shows numerical values of the G_(os) in FRA for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V);

FIG. 4 shows numerical values of the s_(p) for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V);

FIG. 5 shows numerical values of the r_(ep) for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V);

FIG. 6 is an FRA plot for the simulated probe test with k_(H)/k_(V)=10;

FIG. 7 is a plot of pressure changes and pressure derivatives for buildup data;

FIG. 8 is a flow chart illustrating one embodiment of the present invention for determining horizontal and vertical permeabilities from buildup and FRA analysis;

FIG. 9 is a comparison of simulated pressure data with an analytical spherical solution derived using the buildup permeability and an isotropic skin factor;

FIGS. 10 a, 10 b shows use of a probe for two measurements in a near horizontal borehole;

FIG. 11 shows K values for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V);

DETAILED DESCRIPTION OF THE INVENTION

Referring now to FIG. 1, there is illustrated schematically a section of a borehole 10 penetrating a portion of the earth formations 11, shown in vertical section. Disposed within the borehole 10 by means of a cable or wireline 12 is a sampling and measuring instrument 13. The sampling and measuring instrument is comprised of a hydraulic power system 14, a fluid sample storage section 15 and a sampling mechanism section 16. Sampling mechanism section 16 includes selectively extensible well engaging pad member 17, a selectively extensible fluid admitting sampling probe member 18 and bi-directional pumping member 19. The pumping member 19 could also be located above the sampling probe member 18 if desired.

In operation, sampling and measuring instrument 13 is positioned within borehole 10 by winding or unwinding cable 12 from a hoist 19 around which cable 12 is spooled. Depth information from depth indicator 20 is coupled to processor 21. The processor analyzes the measurements made by the downhole tool. In one embodiment of the invention, some or all of the processing may be done with a downhole processor (not shown). A satellite link 23 may be provided to send the data to a remote location for processing.

For any formation testing tool, the flow measurement using a single probe is the cheapest and quickest way. The present invention provides two practical methods to estimate horizontal and vertical permeabilities from such probe test data. The first method is to combine the results of the two analyses, FRA and pressure buildup analysis. The second method is to combine the results of buildup analysis and pressure history matching. The probe test can be conducted using Baker Atlas's formation testing tool used under the service mark RCI^(SM). Some details of the formation testing tool are described in U.S. Pat. No. 5,377,755 issued to Michaels et al., having the same assignee as the present invention and the contents of which are incorporated herein by reference.

The method of the present invention uses data from a drawdown test and a pressure buildup test made with a single probe. The relationship between measured pressure and formation flow rate can be observed in the graph in FIG. 2. The pressure and flow rate measurements are shown as individual points connected by a curve 70. A linear regression analysis of the points on curve 70 can be used to generate a line 72 for which the slope can be calculated. The slope of line 72 is related to the fluid mobility.

As discussed in Sheng et al., if the non-spherical flow pattern is described using a geometric skin factor, s_(p), the spherical drawdown solution may be written

$\begin{matrix} {{{p_{i} - {p(t)}} = {{\frac{q\;\mu}{4\pi\; k_{s}r_{p}}\left( {1 + s_{p}} \right)} - {\frac{q\;\mu}{4\pi\; k_{s}}\sqrt{\frac{\phi\;\mu\; c_{t}}{\pi\; k_{s}}}\frac{1}{\sqrt{t}}}}},} & (2) \end{matrix}$ where

-   c_(t) is the total formation compressibility, atm⁻¹; -   k_(s) is the spherical permeability, D; -   p(t) represents the measured pressure in the tool, atm; -   P_(i) is the initial formation pressure, atm; -   q is the volumetric flow rate, cm³/s; -   r_(p) is the true probe radius, cm; -   s_(p) is the geometric skin factor, dimensionless; -   t is the time since the start of drawdown, s; -   μ is the viscosity of fluid, cP; and -   ∅ is the formation porosity, fraction.     The units of measurement are not relevant except as far as they     pertain to specific numerical values derived later in this document.

The steady-state pressure drop for a single probe in an anisotropic formation was investigated by Dussan and Sharma (1992). On the basis that most of the pressure drop occurs in the vicinity of the probe and the probe is very small in relation to the wellbore, they treated the wellbore as being infinite in diameter (r_(w)=∞). Their pressure drop is formulated by

$\begin{matrix} {{{\Delta\;{p\left( {\eta,r_{p},{r_{w} = \infty}} \right)}} = {\frac{q\;\mu}{2\pi\sqrt{k_{H}k_{V}}{\max\left( {r_{p},{r_{p}/\eta}} \right)}}{F\left( {\frac{\pi}{2},\sqrt{1 - \eta^{2}}} \right)}}},} & (3) \end{matrix}$ where η=k_(V)/k_(h), and F(π/2, e) is the complete elliptical integral of the first kind defined as

$\begin{matrix} {{F\left( {\frac{\pi}{2},e} \right)} = {\int_{0}^{1}{\frac{\mathbb{d}v}{\sqrt{\left( {1 - v^{2}} \right)\left( {1 - {{\mathbb{e}}^{2}v^{2}}} \right)}}.}}} & (4) \end{matrix}$ Note that F tends to π/2 as e defined as √{square root over (1−η²)} tends to zero in an isotropic case.

Wilkinson and Hammond (1990) extended Dussan and Sharma's work to include a correction for the borehole radius by introducing a shape factor, C_(eff). The shape factor is defined as

$\begin{matrix} {{C_{eff}\left( {\eta,r_{p},r_{w}} \right)} = {\frac{\Delta\;{p\left( {\eta,r_{p},r_{w}} \right)}}{\Delta\;{p\left( {\eta,r_{p},{r_{w} = \infty}} \right)}}.}} & (5) \end{matrix}$ Then the pressure drop is

$\begin{matrix} {{{\Delta\;{p\left( {\eta,r_{p},r_{w}} \right)}} = \frac{q\;\mu\;{F\left( {\frac{\pi}{2},\sqrt{1 - \eta^{2}}} \right)}C_{eff}}{2\pi\sqrt{k_{H}k_{V}}{\max\left( {r_{p},{r_{w}/\eta}} \right)}}},{where}} & (6) \\ {C_{eff} = {1 - {{\frac{\max\left( {r_{p},{r_{w}/\eta}} \right)}{4r_{w}{F\left( {\frac{\pi}{2},\sqrt{1 - \eta^{2}}} \right)}}\left\lbrack {3.3417 + {\ln\left( {\frac{r_{w}\eta}{2{r_{p}\left( {1 + \eta} \right)}} - \frac{1}{1 + \eta}} \right)}} \right\rbrack}.}}} & (7) \end{matrix}$ When the wellbore radius tends to infinity, Ceff tends to 1, and eqn. 6 becomes identical to eqn. 3, as should be the case. In the FRA formulation, non-spherical flow geometry is considered by introducing a geometric factor, G₀. The pressure drop induced by a flow rate is

$\begin{matrix} {{{\Delta\;{p\left( {\eta,r_{p},r_{w}} \right)}} = \frac{q\;\mu}{G_{0}k_{FRA}r_{p}}},} & (8) \end{matrix}$ where k_(FRA) is the permeability estimated from the FRA technique.

By comparing eqns. 6 and 8, the values of G_(o)can be derived from the values of F and C_(eff) using the following equation

$\begin{matrix} {G_{0} = {\frac{2\pi\sqrt{k_{H}k_{V}}{\max\left( {r_{p},{r_{p}/\eta}} \right)}}{{F\left( {\frac{\pi}{2},\sqrt{1 - \eta^{2}}} \right)}C_{eff}r_{p}k_{FRA}}.}} & (9) \end{matrix}$ Deriving the values of G₀ using the above equation depends on which permeability k_(FRA) is (horizontal, vertical or spherical permeability). It also involves the calculation of the complete elliptical integral. Sometimes such calculation may not be performed easily, especially when η² is greater than one. It is also found that the values of C_(eff) calculated using eqn. 7 are even larger than 1.0 in the cases of high k_(H)/k_(V), which violates the fluid flow physics. This is attributable to violation of one of the assumptions used in the derivation of eqn 7 when k_(H)/k_(V) is very large. In other words, eqn. 7 is not applicable in some cases. As a result, we may not be able to use eqn. 6 to calculate the pressure drop in some cases.

Wilkinson and Hammond (1990) corrected the values of C_(eff) in the cases of high k_(H)/k_(V). Based on the corrected C_(eff), they defined another parameter, k_(H)/k_(D). Here k_(H) is horizontal permeability, and k_(D) is a drawdown permeability, defined as

$\begin{matrix} {k_{D} = {\frac{q\;\mu}{4r_{p}\Delta\; p}.}} & (10) \end{matrix}$ k_(D) is computed as if the flow occurs in an isotropic formation and the borehole is infinite. In this case, the flow is a hemi-spherical flow, and the equivalent probe radius is 2r_(p)/π. When eqn. 10 is used in an isotropic formation with an infinite wellbore, the estimated k_(D) is the true formation permeability. When eqn. 10 is used in a real anisotropic formation with a real finite wellbore, the estimated k_(D) may not represent the horizontal, vertical, or spherical permeability and is a function of k_(H)/k_(V) and r_(p)/r_(w). Because k_(D) is a function of k_(H)/k_(V) and r_(p)/r_(w), we can use its values to derive the values for other geometric correction factors at different k_(H)/k_(V) and r_(p)/r_(w).

To estimate G_(o), we compare eqn. 8 with eqn. 10, and get

$\begin{matrix} {{G_{0}k_{FRA}} = {{4k_{D}} = {\frac{q\;\mu}{4r_{p}\Delta\; p}.}}} & (11) \end{matrix}$ We note for a particular test that with the measured q and Δp, and the fixed μ, r_(p), the product, G_(o)k_(FRA), is fixed. G_(o) and k_(FRA) are related to each other by the relationship described by the above equation. In other words, depending on the type of permeability sought (e.g., horizontal, vertical, or spherical permeability), different values of G_(o) are required. From eqn. 11,

$\begin{matrix} {G_{0} = {\frac{4k_{D}}{k_{FRA}}.}} & (12) \end{matrix}$ If a spherical permeability from FRA is sought, then it is necessary to use the G_(os) which corresponds to spherical permeability:

$\begin{matrix} {G_{os} = {\frac{4}{k_{s}/k_{D}} = {\frac{4}{\left( {k_{H}^{2/3}{k_{V}^{1/3}/k_{D}}} \right)} = {\frac{4}{k_{H}/k_{D}}{\left( \frac{k_{H}}{k_{V}} \right)^{1/3}.}}}}} & (13) \end{matrix}$ Here spherical permeability has been assumed to be given by k_(s)=√{square root over (k_(V)k_(H) ²)}. From the published values of k_(H)/k_(D) (Wilkinson and Hammond, 1990), the values of G_(os) are readily obtained. The values as a function of r_(p)/r_(w) and k_(H)/k_(V) are tabulated in Table 1 and shown in FIG. 3.

TABLE 1 Numerical values of G_(os) in FRA for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V) r_(p)/r_(w) = k_(H)/k_(V) 0.025 0.05 0.1 0.2 0.3 0.01 3.75 3.75 3.75 3.92 3.92 0.1 3.64 3.64 3.71 3.87 3.95 1 4.08 4.17 4.26 4.44 4.65 10 5.42 5.56 5.78 6.11 6.38 100 8.33 8.60 9.06 9.72 10.26 1000 14.18 14.87 15.81 17.09 18.02 10000 25.96 27.45 29.31 31.57 33.15 100000 49.64 52.60 55.92 59.89 62.51 1000000 97.09 102.30 108.40 115.27 119.76 From Table 1 and FIG. 3, we see that the geometric factor is a strong function of anisotropy and a weak function of r_(p)/r_(w). Also, the values of G_(os) for k_(H)/k_(V) from 1 to 100 calculated from eqn. 13 are in close agreement with those calculated from eqn. 9 in which C_(eff) is calculated using eqn. 7.

The concept of geometric skin was proposed to represent the above defined geometric factor (Strauss, 2002). Defining a geometric skin factor s_(p) to account for the deviation from the true spherical flow gives

$\begin{matrix} {{\Delta\; p} = {\frac{q\;{\mu\left( {1 + s_{p}} \right)}}{4\pi\; k_{s}r_{p}}.}} & (14) \end{matrix}$ Comparing eqn. 10 and eqn. 14, s_(p) can be estimated from

$\begin{matrix} {s_{p} = {\frac{\pi\left( {k_{H}/k_{V}} \right)}{\left( {k_{H}/k_{V}} \right)^{1/3}} - 1.}} & (15) \end{matrix}$ Again from the published values of k_(H)/k_(D) (Wilkinson and Hammond, 1990), the values of s_(p) are readily obtained. The values as a function of r_(p)/r_(w) and k_(H)/k_(D) are tabulated in Table 2 and shown in FIG. 4.

TABLE 2 Numerical values of s_(p) for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V) r_(p)/r_(w) = k_(H)/k_(V) 0.025 0.05 0.1 0.2 0.3 0.01 2.35 2.35 2.35 2.21 2.21 0.1 2.45 2.45 2.38 2.25 2.18 1 2.08 2.02 1.95 1.83 1.70 10 1.32 1.26 1.17 1.06 0.97 100 0.51 0.46 0.39 0.29 0.23 1000 −0.11 −0.15 −0.21 −0.26 −0.30 10000 −0.52 −0.54 −0.57 −0.60 −0.62 100000 −0.75 −0.76 −0.78 −0.79 −0.80 1000000 −0.87 −0.88 −0.88 −0.89 −0.90 Again, there is little dependence on the probe packer size as measured by the dimensionless probe size, r_(p)/r_(w).

If we define an equivalent probe radius, r_(ep), to account for the deviation from true spherical flow, we can write

$\begin{matrix} {{\Delta\; p} = {\frac{q\;\mu}{4\pi\; k_{s}r_{sp}}.}} & (16) \end{matrix}$ Comparing eqns. 10 and 16, r_(ep) can be estimated from

$\begin{matrix} {r_{ep} = {\frac{{r_{p}\left( {k_{H}/k_{V}} \right)}^{1/3}}{\pi\left( {k_{H}/k_{D}} \right)}.}} & (17) \end{matrix}$

Using the published values of k_(H)/k_(D) (Wilkinson and Hammond, 1990), the values of r_(ep) are readily obtained. The values as a function of r_(p)/r_(w) and k_(H)/k_(V) are tabulated in Table 3 and shown in FIG. 5.

TABLE 3 Numerical values of the r_(ep)/r_(p) for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V) r_(p)/r_(w) = k_(H)/k_(V) 0.025 0.05 0.1 0.2 0.3 0.01 0.30 0.30 0.30 0.31 0.31 0.1 0.29 0.29 0.30 0.31 0.31 1 0.32 0.33 0.34 0.35 0.37 10 0.43 0.44 0.46 0.49 0.51 100 0.66 0.68 0.72 0.77 0.82 1000 1.13 1.18 1.26 1.36 1.43 10000 2.07 2.18 2.33 2.51 2.64 100000 3.95 4.19 4.45 4.77 4.97 1000000 7.73 8.14 8.63 9.17 9.53 The formulation and values of the above correction factors are based on the related spherical flow eqns. 8, 14, or 16. For eqn. 6, k_(FRA) is assumed to be the spherical permeability. Comparing their defining eqns. 11, 13, and 15, it can be seen that these three correction factors have the following relationship:

$\begin{matrix} {{{{s_{p} + 1} = {\frac{4\pi}{G_{os}} = \frac{1}{r_{ep}/r_{p}}}};}{or}} & (18) \\ {{{s_{p} + 1} = \frac{4\pi}{G_{os}}},} & (19) \\ {\frac{1}{r_{ep}/r_{p}} = {\frac{4\pi}{G_{os}}.}} & \left( {19a} \right) \end{matrix}$

Substituting from eqn. 19 into eqn. 2, gives:

$\begin{matrix} {{p_{i} - {p(t)}} = {\frac{q\;\mu}{G_{os}k_{s}r_{p}} - {\frac{q\;\mu}{4\pi\; k_{s}}\sqrt{\frac{\phi\;\mu\; c_{t}}{\pi\; k_{s}}}{\frac{1}{\sqrt{t}}.}}}} & (20) \end{matrix}$ Eqns. 2 and 20 are valid for both isotropic and anisotropic formations. Using the principle of superposition, the buildup solution is

$\begin{matrix} {{{p(t)} = {p_{i} - {\frac{q\;\mu}{4\pi\; k_{s}}\sqrt{\frac{\phi\;\mu\; c_{t}}{\pi\; k_{s}}}\left( {\frac{1}{\sqrt{\Delta\; t}} - \frac{1}{\sqrt{t}}} \right)}}},} & (21) \end{matrix}$ where Δt is the shut-in time, s. Here, q is the flow rate for the previous drawdown measurement. According to eqn. 20, the buildup mobility is estimated from

$\begin{matrix} {{\left( \frac{k_{s}}{\mu} \right)_{BU} = {\frac{1}{\pi}\left( \frac{q}{4m_{s}} \right)^{2/3}\left( {\phi\; c_{t}} \right)^{1/3}}},} & (22) \end{matrix}$ where m_(s), is the slope of the linear plot of p(t) vs. the time function (Δt^(−1/2)−t^(−1/2)). For the purposes of the present invention, the permeability measured using the buildup measurements is referred to as a first permeability.

Turning now to the FRA method as described in Kasap,

$\begin{matrix} {{{p(t)} = {p_{i} - \frac{q_{f}\mu}{k_{s}G_{os}r_{p}}}},} & (23) \end{matrix}$ where q_(f), the formation flow rate at the sand face near the probe, is

$\begin{matrix} {{q_{f} = {{c_{sys}V_{sys}\frac{\mathbb{d}{p(t)}}{\mathbb{d}t}} + q_{dd}}},} & (24) \end{matrix}$ corrected for the storage effect. In the above equation, C_(sys) is the compressibility of the fluid in the tool, atm⁻¹; q_(dd) is the piston withdrawal rate, cm³/s ; V_(sys) is the system (flow line) volume, cm³.

According to FRA, the data in both drawdown and buildup periods are combined to estimate the mobility from

$\begin{matrix} {{\left( \frac{k_{s}}{\mu} \right)_{FRA} = \frac{1}{G_{os}r_{p}m_{FRA}}},} & (25) \end{matrix}$ where m_(FRA) is the slope of the linear plot of p(t) vs. q_(f). By plotting the drawdown data and the buildup data in the FRA plot (FIG. 2), if both data are seen to fall on the same straight line with a slope, m_(FRA), the estimated permeability from the drawdown and the buildup is the same. That means within the radius of investigation for the drawdown and buildup, the formation is homogeneous. This is the condition for the presented methods to work.

Eqn. 25 shows that the estimated mobility from FRA is affected by the local flow geometry indicated by G_(os). Thus, a correct value of G_(os) must be provided. However, G_(os) strongly depends on the ratio of vertical-to-horizontal permeability that is generally unknown before the test is performed. In this case, the value of G_(os) in an isotropic formation is used. As a result, the FRA estimated permeability may not represent the true spherical permeability. In contrast, the spherical permeability can be obtained from a buildup analysis without prior knowledge of formation anisotropy, and the estimate of mobility from the buildup analysis is not affected by the local flow geometry according to eqn. 22. In other words, the correct estimate of spherical permeability can be obtained from buildup analysis without knowing formation anisotropy and local flow geometry. The difference in the estimated spherical permeability from buildup analysis and FRA, discussed in the above, can be used to estimate the horizontal and vertical permeabilities. For the purposes of the present invention, the permeability determined by FRA processing is referred to as a second permeability.

The difference in the estimated spherical permeability from buildup analysis (the first permeability) and FRA permeability (the second permeability), discussed in the above, can be used to estimate the horizontal and vertical permeabilities. A simulated probe-pressure test data as an example is used to illustrate the procedures. First, the probe-pressure test simulation is described.

The simulation model used is given in Table 4.

TABLE 4 Input parameters used in simulation Porosity, fraction 0.2 Spherical permeability, mD 10 k_(H)/k_(V) 10 Viscosity, cP 1 Formation pressure, psi 4000 Fluid compressibility, 1/psi 2.50E−06 Wellbore radius, cm. 6.35 Probe radius, cm 0.635 Flow line volume, ml 371 Drawdown rate, ml/s 4 Duration of drawdown, s 10 The symmetry in the problem is used to reduce the model to one quarter of the probe and the formation. Further, the effect of gravity is neglected. The model is a radial model. The k_(H)/k_(V) is equal to 10 with the spherical permeability of 10 mD. The r_(p)/r_(w) is equal to 0.1. The drawdown rate for the quarter model is 1 ml/s.

Next, results of analyzing the simulation data using the FRA technique are discussed. FIG. 6 shows the expected linear relation between the pressure and the formation flow rate. If the data were real probe-pressure test data and k_(H)/k_(V) were unknown, one could logically assume the formation were isotropic. According to Table 1, the geometric factor, G_(os), would be 4.26 for r_(p)/r_(w) equal to 0.1. Based on eqn. 25 and using this value of G_(os), one would estimate a spherical permeability of 13 mD.

The simulated pressure test data could also be analyzed using buildup (BU) analysis using any pressure transient analysis software with spherical flow solutions. For this example, the commercially available software Interpret2003 of Paradigm Geophysical Co was used. FIG. 7 shows the buildup analysis plot to estimate the spherical permeability for this case. The abscissa is time and ordinate is the pressure change 701 or the pressure derivative 703. The plot is on a log-log scale. Also shown on the plot are lines with a slop of +1 (705) and a slope of −½ (707). The spherical flow regime is identified by a negative half slope in the log-log derivative plot. From this buildup analysis, the spherical permeability is estimated to be 9.62 mD, close to the input spherical permeability. It should be noted that the use of the Interpret2003 software is for exemplary purposes only and other software packages that perform similar functions (as described below) could be used.

For the same pressure data, different estimates of permeability are obtained from buildup analysis and from FRA. One is 13 mD from FRA, the other is 9.62 mD from the BU analysis. The latter is close to the actual permeability used in the simulation model. The former is different from the actual permeability because we used an incorrect G_(os). To make FRA estimated permeability closer to the actual one used in the simulation, a value of G_(os) appropriate for the permeability anisotropy ratio in the simulation should be used. Assuming the BU estimated spherical permeability is correct, the correct G_(os) can be estimated as follows.

$\begin{matrix} {{\left( {k_{s}G_{os}} \right)_{FRA} = \frac{\mu}{r_{p}m_{FRA}}},} & (26) \end{matrix}$ The above shows that for a particular test, since the linear relationship between the measured q and Δp results in a constant slope, m_(FRA), for the fixed μ and r_(p), the product, (G_(os)k_(s))FRA, is fixed. In other words, for a particular test, if an isotropic formation is assumed for FRA, then (G_(os)k_(s)) in the isotropic formation, denoted by (G_(os)k_(s))_(iso), should equal the permeability-geometric factor product of the anisotropic formation, (G_(os)k_(s))_(ani). This product consists of the correct G_(os) and the correct k_(s) in the anisotropic formation. Because the BU estimated permeability is assumed to be the true spherical permeability, then the correct G_(os) in the anisotropic formation, (G_(o))_(ani), can be estimated from

$\begin{matrix} {\left( G_{os} \right)_{ani} = {\frac{\left( {G_{os}k_{s}} \right)_{iso}}{\left( k_{s} \right)_{BU}}.}} & (27) \end{matrix}$

In the term (G_(os)k_(s))_(iso) of the above equation, G_(os) is the geometric factor for an isotropic formation (G_(os)=4.26 from Table 1), and k_(s) is the FRA permeability estimated initially assuming the formation is isotropic. For this example, k_(s) is 13 mD. In the denominator, (k_(s))_(BU) is the spherical permeability estimated from the buildup analysis which is 9.62 mD in this example. Therefore, the correct G_(os) in this example is

$\begin{matrix} {\left( G_{os} \right)_{ani} = {\frac{\left( {G_{os}k_{s}} \right)_{iso}}{\left( k_{s} \right)_{BU}} = {\frac{(4.26)(13)}{9.62} = {5.76.}}}} & (28) \end{matrix}$ The estimated G_(os) of 5.76 is very close to the G_(os) in FIG. 1 when k_(H)/k_(V) is equal to 10 and r_(p)/r_(w) is equal to 0.1. Therefore, by combining the results of FRA and buildup analysis, it is possible to determine k_(H)/k_(V). Having k_(H)/k_(V) determined, the horizontal permeability and vertical permeability are readily obtained: k _(H)=(k _(s))_(BU)/(k _(H)/k _(V))^((1/3)),  (29), k _(V)=k _(H)/(k _(H)/k _(V)).  (30).

For this example, the calculated horizontal permeability and vertical permeability are 20.7 mD and 2.07 mD, respectively. These values are very close to their respective simulation model input values of 21.54 mD and 2.15 mD. Thus the method to combine FRA and buildup analysis is demonstrated.

FIG. 8 is a flow chart illustrating the first embodiment of the invention. Pressure buildup data 751 are analyzed to get a first estimate of spherical permeability. Separately, the pressure buildup data 751 and the drawdown data 757 are analyzed to get a second estimate of spherical permeability 759. Using the two different permeabilities, the geometric factor G_(os) for the probe is corrected 755 and using the corrected G_(os), the horizontal and vertical permeabilities are determined as discussed above.

A second embodiment of the present invention uses the spherical permeability obtained from the pressure buildup test (the first permeability) as a starting point for matching the entire pressure history, including the drawdown data. In Interpret2003 the geometric skin factor, s_(p), is used to describe the non-spherical flow near the probe. Even though the local geometry near the probe does not affect the permeability estimate, it does affect the pressure data as given by eqn. 2. In the above example, using the BU estimated permeability of 9.62 mD and an isotropic geometric skin factor of 1.95 shown in Table 1, the pressure data from Interpret2003 cannot be matched with the simulated pressure data because we used the wrong isotropic geometric skin factor. This is shown in FIG. 9 where the abscissa is time and the ordinate is pressure. The buildup portion is used to derive the permeability and this derived permeability is used to model the pressure data. More obviously, the modeled drawdown data 803 does not match the actual drawdown data 801. To match the simulated pressure data, it is necessary to use the BU estimated spherical permeability, and also to change the value of s_(p) until the pressure data from Interpret2003 matches the numerical simulation data. It is found that using a value of s_(p) equal to 1.2, a good match is obtained (not shown). From Table 2 it can be seen that s_(p) equal to 1.2 (close the s_(p) of 1.17 in Table 1) corresponds to k_(H)/k_(V) equal to 10 and r_(p)/r_(w) equal to 0.1. As above, k_(H)/k_(V) has been estimated to be equal to 10. Once k_(H)/k_(V) is obtained, eqns 28 and 29 can be used to estimate the horizontal and vertical permeabilities. Thus, the second method also uses a permeability from BU analysis (the first method) in combination with matching the entire pressure data (processing of data over the entire time interval including drawdown and buildup) to estimate horizontal and vertical permeabilities.

Conceptually, the second method is based on deriving a spherical permeability based on a buildup analysis, and then using this determined spherical permeability to match the pressure history data by adjusting the geometric skin factor. Knowledge of the spherical permeability and the geometric skin factor makes it possible to determine the horizontal and vertical permeabilities.

The two embodiments of the present invention discussed above are used to estimate horizontal and vertical permeabilities based on the assumption of a homogeneous and anisotropic formation. Such an assumption is reasonable in a practical probe test, because the formation on the small scale near the probe probably can be considered virtually homogeneous. Therefore, the invention provides a way to estimate the horizontal and vertical permeabilities from a single probe test without additional information. This is in contrast to prior art methods that require simultaneous measurements with multiple probes, or measurements with a specially designed probe in two orientations.

In another embodiment of the invention, two tests are made in a near horizontal borehole. In one test, the probe is set and sealed horizontally against a side wall of the borehole. This is schematically illustrated in FIG. 10 a wherein the borehole 851 is shown in cross-section and a probe 853 is in contact with the side wall of the borehole. In a second test, schematically illustrated in FIG. 10 b, the probe 853 is shown against the upper wall of the borehole. It is to be noted that the method is equally applicable if, in the second test, the probe is against the bottom wall of the borehole.

The solution for the first test is the same as that in a vertical well, and has been discussed above. The solution for the second test is derived next. The objective is to determine the relationship between the pressure at the probe and the fluid withdrawal rate from the anisotropic formation. As before, a cylindrical coordinate system is used in which the wellbore wall near the probe can be approximated by the z=0 plane, with the formation located in the half-space z≧0. The initial formation pressure is p_(i). The z axis for the test of FIG. 10 b coincides with the vertical direction. The perimeter of the probe opening through which fluid flows is given by r²=r_(p) ² at z=0. The flowing pressure at the probe opening is p_(p). There is no flow across the rest of the plane at z=0. The mathematical description of such probe test is a mixed boundary problem. Its formulation is given as follows.

$\begin{matrix} {{{{k_{H}\left( {\frac{\partial^{2}p}{\partial r^{2}} + {\frac{1}{r}\frac{\partial p}{\partial r}}} \right)} + {k_{V}\frac{\partial^{2}p}{\partial z^{2}}}} = 0},} & (31) \\ {{p = {{{p_{p}\mspace{14mu}{at}\mspace{14mu} r} \leq {r_{p}\mspace{14mu}{and}\mspace{14mu} z}} = 0}},} & (32) \\ {{\frac{\partial p}{\partial z} = {{{0\mspace{14mu}{at}\mspace{14mu} r} > {r_{p}\mspace{14mu}{and}\mspace{14mu} z}} = 0}},} & (33) \\ {\left. p\rightarrow\left. {{p_{i}\mspace{14mu}{as}\mspace{14mu} r^{2}} + z^{2}}\rightarrow{{\infty\mspace{14mu}{and}\mspace{14mu} z} \geq 0} \right. \right.,} & (34) \end{matrix}$

Of interest is the relationship between pressure drop, p_(i)-p_(p), and flow rate, q. This is done by evaluating the integral:

$\begin{matrix} {q = \left. {\frac{2\pi\; k_{V}}{\mu}{\int_{A_{p}}^{\;}\frac{\partial p}{\partial z}}} \middle| {}_{z = 0}{r{{\mathbb{d}r}.}} \right.} & (35) \end{matrix}$ In the above equations,

-   A_(p) represents area of probe opening, cm² -   k_(H) represents horizontal permeability, D -   k_(V) represents vertical permeability, D -   p represents pressure, atm -   p_(i) represents initial formation pressure, atm -   p_(p) represents pressure at the probe, atm -   q represents volumetric flow rate, cm³/s -   r represents radial coordinate of cylindrical grid system, cm -   r_(p) represents true probe radius, cm -   z represents z axis in the coordinate system, cm -   μ represents viscosity of fluid, cP     The units of measurement are not relevant except as far as they are     consistently follow one unit system. Here Darcy unit system is used.

Using the following notation:

$\begin{matrix} {{r^{\prime} = r},} & (36) \\ {{z^{\prime} = {\sqrt{\frac{k_{H}}{k_{V}}}z}},} & (37) \end{matrix}$ the above mathematical formulation (Eqns. 31 to 35) is converted in the following formulation:

$\begin{matrix} {{{\left( {\frac{\partial^{2}p}{\partial r^{\prime\; 2}} + {\frac{1}{r^{\prime}}\frac{\partial p}{\partial r^{\prime}}}} \right) + \frac{\partial^{2}p}{\partial z^{\prime\; 2}}} = 0},} & \left( 31^{\prime} \right) \\ {{p = {{{p_{p}\mspace{14mu}{at}\mspace{14mu} r^{\prime}} \leq {r_{p}\mspace{14mu}{and}\mspace{14mu} z^{\prime}}} = 0}},} & \left( 32^{\prime} \right) \\ {{\frac{\partial p}{\partial z^{\prime}} = {{{0\mspace{14mu}{at}\mspace{14mu} r^{\prime}} > {r_{p}\mspace{14mu}{and}\mspace{14mu} z^{\prime}}} = 0}},} & \left( 33^{\prime} \right) \\ {\left. p\rightarrow\left. {{p_{i}\mspace{14mu}{as}\mspace{14mu} r^{\prime 2}} + \frac{z^{\prime 2}}{k_{H}/k_{V}}}\rightarrow{{\infty\mspace{14mu}{and}\mspace{14mu} z^{\prime}} \geq 0} \right. \right.,} & \left( 34^{\prime} \right) \\ {q = \left. {\frac{2\pi\;\sqrt{k_{H}k_{V}}}{\mu}{\int_{A_{p}}^{\;}\frac{\partial p}{\partial z^{\prime}}}} \middle| {}_{z^{\prime} = 0}{r^{\prime}{{\mathbb{d}r^{\prime}}.}} \right.} & \left( 35^{\prime} \right) \end{matrix}$

The solution for the above problem was solved by Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Oxford University Press (1959). According to their solution, the relationship between pressure drop and flow rate for the above problem is

$\begin{matrix} {q = {\frac{4\sqrt{k_{H}k_{V}}{r_{p}\left( {p_{i} - p_{p}} \right)}}{\mu}.}} & (38) \end{matrix}$ Note that from the above equation, it is possible to obtain a permeability (k_(H)k_(V))^(1/2), a geometric average permeability of horizontal permeability and vertical permeability.

For the first test with the probe set horizontally against the side wall (FIG. 10 a) in a horizontal well, the relationship between the pressure drop and flow rate is the same as that in a vertical well. Using the geometric factor and horizontal permeability, the relationship derived above is

$\begin{matrix} {{{p_{i} - p_{p}} = \frac{q\;\mu}{G_{oH}k_{H}r_{p}}},} & (39) \end{matrix}$ where G_(oH) is the geometric factor when the pressure drop vs. flow rate relationship is formulated using horizontal permeability, k_(H). Its values at different k_(H)/k_(V) and r_(p)/r_(w) are reported in the same reference and reprinted here in Table 5. Here r_(w) is the radius of wellbore. Note that the values in Table 5 are for G_(0H), related to a horizontal permeability whereas the values in Table 1 are for G_(0S), related to a spherical permeability.

TABLE 5 Numerical values of G_(oH) (for k_(H)) for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V) r_(p)/r_(w) = k_(H)/k_(V) 0.025 0.05 0.1 0.2 0.3 0.01 17.39 17.39 17.39 18.18 18.18 0.1 7.84 7.84 8.00 8.33 8.51 1 4.08 4.17 4.26 4.44 4.65 10 2.52 2.58 2.68 2.84 2.96 100 1.79 1.85 1.95 2.09 2.21 1000 1.42 1.49 1.58 1.71 1.80 10000 1.20 1.27 1.36 1.47 1.54 100000 1.07 1.13 1.20 1.29 1.35 1000000 0.97 1.02 1.08 1.15 1.20

From Eqn. 39, the horizontal permeability can be obtained. But this permeability is closely related to the geometric factor which is a strong function of k_(H)/k_(V). Before analyzing the test data, k_(H)/k_(V) is unknown. However, for a particular test with the measured q and p_(p), and the fixed μ, r_(p), the product G_(oH)k_(H) is a determined quantity. For the second test in a horizontal well when the probe is set vertically against the top wall of the borehole (FIG. 10 b), the relationship between the pressure drop and flow rate is describe by Eqn. 38 and a mean permeability, (k_(H)k_(V))^(1/2) can be obtained. In other words, when the two tests are conducted at the same measured depth, the following two quantities are obtained:

$\begin{matrix} {{{K_{S} \equiv {G_{oH}k_{H}}} = \frac{q_{S}\mu}{r_{p}\left( {p_{i} - p_{p,S}} \right)}},} & (40) \\ {{{K_{T} \equiv \sqrt{k_{H}k_{V}}} = \frac{q_{T}\mu}{4{r_{p}\left( {p_{i} - p_{p,T}} \right)}}},} & (41) \end{matrix}$ where the subscripts S and T means the probe is set horizontally against the side wall and vertically against the top wall, respectively. Both K_(S) and K_(T) are functions of permeability anisotropy, k_(H)/k_(V). Now we define another quantity K using these two quantities:

$\begin{matrix} {{K \equiv \frac{K_{S}}{K_{T}}} = {G_{o\; H}{\sqrt{\frac{k_{H}}{k_{V}}}.}}} & (12) \end{matrix}$

Because G_(oH) is a function of k_(H)/k_(V), K is also a function of k_(H)/k_(V). Using the values of G_(oH) in Table 4, the values of K are obtained as shown in Table 6 and FIG. 11 as a function of r_(p)/r_(w) and k_(H)/k_(V). For the two pretests conducted at the same measured depth, the K value can be calculated using K_(S) and K_(T) from Eqns. 10 and 11. Then the k_(H)/k_(V) at the measured depth can be obtained by looking up Table 6 or FIG. 11 using the calculated K value and the known value of r_(p)/r_(w). From knowledge of k_(H)/k_(V), the horizontal and vertical permeabilities are readily determined:

$\begin{matrix} {{k_{H} = {K_{T}\sqrt{\frac{k_{H}}{k_{V}}}}},} & (43) \\ {k_{V} = {\frac{k_{H}}{\left( {k_{H}/k_{V}} \right)}.}} & (44) \end{matrix}$

TABLE 6 Numerical values of K for various values of r_(p)/r_(w) and anisotropy k_(H)/k_(V) r_(p)/r_(w) = k_(H)/k_(V) 0.025 0.05 0.1 0.2 0.3 0.01 1.74 1.74 1.74 1.82 1.82 0.1 2.48 2.48 2.53 2.64 2.69 1 4.08 4.17 4.26 4.44 4.65 10 7.96 8.16 8.49 8.97 9.37 100 17.94 18.52 19.51 20.94 22.10 1000 44.86 47.02 50.00 54.06 56.98 10000 120.48 127.39 136.05 146.52 153.85 100000 338.21 358.33 381.00 408.04 425.90 1000000 970.87 1023.02 1084.01 1152.74 1197.60

The above equations are derived based on the assumptions of a constant withdrawal rate and steady state flow. In a low permeability formation, the steady state flow condition cannot be satisfied unless a long test time is used. A constant drawdown rate is not reachable in practice because the tool needs time for acceleration and deceleration. The storage effect also makes it difficult to reach a constant rate. In an alternate embodiment of the present invention, both drawdown and buildup tests are made at substantially the same depth with the probe against a sidewall and an upper (or lower) wall. The Formation Rate Analysis (FRA) presented in U.S. Pat. No. 5,708,204 to Kasap, the contents of which are incorporated herein by reference, are used to calculate the above K_(S) and K_(T).

As disclosed in Kasap, pressure is measured with respect to time at a probe in hydraulic communication with the earth formation. Time derivatives of the measured pressure are determined while withdrawing fluid from the earth formation by increasing a volume of a chamber in hydraulic communication with the probe. The volume of the chamber is measured as a function of time and the time derivative of the volume is calculated. This is part of a withdrawal phase. Fluid withdrawal is stopped and the pressure buildup is measured. The permeability is calculated when the measured pressure substantially stops increasing, by determining a slope of a linear relationship of the measured pressure with respect to a fluid flow rate calculated from the time derivative of the volume and the time derivative of the pressure.

The invention has been described in terms of measurements made using logging tools conveyed on a wireline in a borehole. As noted above, The method can also be used on data obtained using measurement-while-drilling sensors on a bottomhole assembly (BHA) conveyed by a drilling tubular. Such a device is described, for example, in U.S. Pat. No. 6,640,908 to Jones et al., and in U.S. Pat. No. 6,672,386 to Krueger et al., having the same assignee as the present invention and the contents of which are fully incorporated herein by reference. The method disclosed in Krueger comprises conveying a tool into a borehole, where the borehole traverses a subterranean formation containing formation fluid under pressure. A probe is extended from the tool to the formation establishing hydraulic communication between the formation and a volume of a chamber in the tool. Fluid is withdrawn from the formation by increasing the volume of the chamber in the tool with a volume control device. Data sets are measured of the pressure of the fluid and the volume of the chamber as a function of time.

The embodiments of the invention that require making measurements on two different walls of a substantially horizontal borehole are readily accomplished in a MWD implementation. If the tests are performed after the well has been drilled, several options are available. One is to convey the pressure tester on coiled tubing. Alternatively, a downhole traction device such as that disclosed in U.S. Pat. No. 6,062,315 to Reinhardt, having the same assignee as the present invention and the contents of which are fully incorporated herein by reference, may be used to convey the pressure tester into the borehole. A traction device may also be used to withdraw the pressure tester from the borehole, or, alternatively, the withdrawal may be done using a wireline.

The processing of the measurements made by the probe in wireline applications may be done by the surface processor 21 or may be done by a downhole processor (not shown). For MWD applications, the processing may be done by a downhole processor that is part of the BHA. This downhole processing reduces the amount of data that has to be telemetered. Alternatively, some or part of the data may be telemetered to the surface. In yet another alternative, the pressure and flow measurements may be stored on a suitable memory device downhole and processed when the drillstring is tripped out of the borehole.

The operation of the probe may be controlled by the downhole processor and/or the surface processor. Implicit in the control and processing of the data is the use of a computer program implemented on a suitable machine readable medium that enables the processor to perform the control and processing. The machine readable medium may include ROMs, EPROMs, EAROMs, Flash Memories and Optical disks.

While the foregoing disclosure is directed to the specific embodiments of the invention, various modifications will be apparent to those skilled in the art. It is intended that all such variations within the scope and spirit of the appended claims be embraced by the foregoing disclosure. 

1. A method of estimating a permeability of an earth formation, the formation containing a formation fluid, the method comprising: (a) withdrawing fluid from the earth formation using a probe having a substantially circular aperture in hydraulic communication with the formation; (b) stopping the withdrawal of the fluid at a defined time; (c) making transient pressure measurements before and after the defined time; (d) obtaining a first estimate of permeability by analyzing a subset of the pressure measurements made only after the defined time and a second estimate of permeability from the pressure measurements made before and after the defined time; and (e) determining from the first and second estimates of permeability at least one of (A) a horizontal permeability of the formation, and (B) a vertical permeability of the formation.
 2. The method of claim 1 further comprising determining at least one of (I) a ratio of a horizontal permeability and a vertical permeability of the formation, and (II) a spherical permeability of the formation.
 3. The method of claim 1 wherein step (e) further comprises using at least one of (i) a geometric factor relating to anisotropy, and (ii) a geometric skin factor relating to non-spherical fluid flow.
 4. The method of claim 1 wherein obtaining the second estimate of permeability further comprises matching pressure measurements made before and after said defined time using the first permeability and by adjusting a geometric skin factor.
 5. The method of claim 1 wherein determining said first permeability further comprises using a relation of the form: ${p(t)} = {p_{i} - {\frac{q\;\mu}{4\pi\; k_{s}}\sqrt{\frac{{\phi\mu}\; c_{t}}{\pi\; k_{s}}}\left( {\frac{1}{\sqrt{\Delta\; t}} - \frac{1}{\sqrt{t}}} \right)}}$ c_(t) is a total formation compressibility; k_(s) is a spherical permeability; p(t) represents a measured pressure in the tool; p_(i) the initial formation pressure; q is a volumetric flow rate; r_(p) is a true probe radius; t is a time after said defined time; μ is a viscosity of fluid; and ∅ is a formation porosity, fraction.
 6. The method of claim 1 further comprising conveying the probe into the borehole on one of (i) a wireline, (ii) a drilling tubular, and (iii) a slickline.
 7. The method of claim 1 wherein the borehole is substantially vertical and the probe is oriented with its axis in any direction.
 8. The method of claim 1 wherein the borehole comprises a deviated borehole.
 9. The method of claim 8 wherein the probe aperture lies substantially in a vertical plane.
 10. An apparatus for evaluating an earth formation, the formation containing a formation fluid, the apparatus comprising: (a) a probe having a substantially circular aperture configured to be conveyed in a borehole and in hydraulic communication with the earth formation; (b) a chamber in communication with the probe, the chamber configured to withdraw fluid from the formation and stopping the withdrawal at a predefined time; (c) a pressure sensing device configured to make transient pressure measurements in said probe; and (d) a processor configured to: (A) estimate a first permeability by analyzing a subset of the pressure measurements made only after the defined time and a second permeability from the pressure measurements made before and after the defined time, and (B) determine from the first and second permeability at least one of (I) a horizontal permeability of the formation, and (II) a vertical permeability of the formation.
 11. The apparatus of claim 10 wherein the processor is further configured to determine at least one of (I) a ratio of a horizontal permeability and a vertical permeability of the formation, and (II) a spherical permeability of the formation.
 12. The apparatus of claim 10 wherein the processor is configured to perform step (B) by using at least one of (i) a geometric factor relating to anisotropy, and (ii) a geometric skin factor relating to non-spherical fluid flow.
 13. The apparatus of claim 10 wherein the processor is configured to obtain the second estimate of permeability at least in part by matching pressure measurements made before and after said defined time using the first permeability and by adjusting a geometric skin factor.
 14. The apparatus of claim 10 wherein said processor is configured to determine the first permeability further using a relation of the form below: ${p(t)} = {p_{i} - {\frac{q\;\mu}{4\pi\; k_{s}}\sqrt{\frac{\phi\;\mu\; c_{t}}{\pi\; k_{s}}}\left( {\frac{1}{\sqrt{\Delta\; t}} - \frac{1}{\sqrt{t}}} \right)}}$ c_(t) is a total formation compressibility; k_(s) is a spherical permeability; p(t) represents a measured pressure in the tool; p_(i) the initial formation pressure; q is a volumetric flow rate; r_(p) is a true probe radius; t is a time after said defined time; μ is a viscosity of fluid; and ∅ is a formation porosity, fraction.
 15. The apparatus of claim 10 further comprising a conveyance device configured to convey the probe into the borehole, the conveyance device selected from (i) a wireline, and (ii) a drilling tubular.
 16. The apparatus of claim 10 wherein the borehole is substantially vertical and the probe is oriented with its axis in any direction.
 17. The apparatus of claim 10 wherein the borehole comprises a deviated borehole.
 18. The apparatus of claim 17 wherein the probe aperture lies substantially in a vertical plane.
 19. A computer readable medium for use with an apparatus for evaluating an earth formation, the formation containing a formation fluid, the apparatus comprising: (a) a probe having a substantially circular aperture configured to be conveyed in a borehole, the probe being in hydraulic communication with the earth formation; (b) a chamber in communication with the probe, the chamber configured to withdraw fluid from the formation and stopping the withdrawal at a predefined time; and (c) a pressure sensing device configured to make transient pressure measurements in said probe; the medium comprising instructions which enable a processor to (d) estimate a first permeability by analyzing a subset of the pressure measurements made only after the defined time and a second permeability from the pressure measurements made before and after the defined time, and (e) determine from the first and second permeability at least one of (I) a horizontal permeability of the formation, and (H) a vertical permeability of the formation.
 20. The medium of claim 19 further comprising at least one of: (i) a ROM, (ii) an EPROM, (iii) an EEPROM, (iv) a Flash Memory, and (v) an optical disk. 